The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras

Abstract

The universal Askey-Wilson algebra $AW(3)$ can be obtained as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1){)}^{\otimes 3}$. We analyze the commutant of ${\mathfrak{o}}_{q^{1/2}}(2)\oplus {\mathfrak{o}}_{q^{1/2}}(2)\oplus {\mathfrak{o}}_{q^{1/2}}(2)$ in $q$-oscillator representations of ${\mathfrak{o}}_{q^{1/2}}(6)$ and show that it also realizes $AW(3)$. These two pictures of $AW(3)$ are shown to be dual in the sense of Howe; this is made clear by highlighting the role of the intermediate Casimir elements of each members of the dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(6))$. We also generalize these results. A higher rank extension of the Askey-Wilson algebra denoted $AW(n)$ can be defined as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1){)}^{\otimes n}$ and a dual description of $AW(n)$ as the commutant of ${\mathfrak{o}}_{q^{1/2}}(2{)}^{\oplus n}$ in $q$-oscillator representations of ${\mathfrak{o}}_{q^{1/2}}(2n)$ is offered by calling upon the dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$.

14 pages